I'm trying to solve the following problem:
Person A has 12 friends: 7 women and 5 men.
Person B has 12 friends: 5 women and 7 men.
How many ways are there to invite 6 men and 6 women to a party, with the condition that there must be 6 friends from person A and 6 friends from person B?
Here is my attempt:
- 6 women who are friends of A and 6 men who are friends of B can be chosen in $ {\binom{7}{6}}^2$ ways because for each way of choosing the 6 women from the 7 female friends of A we have another $\binom{7}{6}$ ways of choosing the 6 men from the 7 male friends of B. So this can be done in $ {\binom{7}{6}}^2 = 49$ ways.
- 6 women who are friends of B and 6 men who are friends of A would be impossible because B has only 5 friends who are women and A has only 5 friends who are men. So in this case we have 0 ways.
- If there are 5 men from A, 1 woman from A, 1 man from B, and 5 women from B we have $\binom{5}{5} * \binom{7}{1} * \binom{7}{1} * \binom{5}{5} = 49$ ways.
- If there are 1 man from A, 5 women from A, 5 men from B, and 1 woman from B we have $\binom{5}{1} * \binom{7}{5} * \binom{7}{5} * \binom{5}{1} = 11025$ ways.
- If there are 4 men from A, 2 women from A, 2 men from B, and 4 women from B we have $\binom{5}{4} * \binom{7}{2} * \binom{7}{2} * \binom{5}{4} = 11025$ ways.
- If there are 2 men from A, 4 women from A, 4 men from B, and 2 women from B we have $\binom{5}{2} * \binom{7}{4} * \binom{7}{4} * \binom{5}{2} = 122500$ ways.
- If there are 3 men from A, 3 women from A, 3 men from B, and 3 women from B we have $\binom{5}{3} * \binom{7}{3} * \binom{7}{3} * \binom{5}{3} = 122500$ ways.
In total there are: $2*49+2*11025+2*122500 = 267148$ ways of inviting 6 men and 6 women with the condition that there are 6 friends from A and 6 friends from B.
Is this correct? If it is not, could someone please explain my mistakes?
And if it is, is there a less 'brute force' solution? (I suspect there is one because some numbers are repeated so maybe there is an identity with the binomial coefficient that I either don't know or don't realize how to apply here)